Calculus Examples

$5 \sec^{2} (\frac{π}{3}) \sin (\frac{π}{3}) + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$5 \cdot 2^{2} \sin (\frac{π}{3}) + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.2

Raise $2$ to the power of $2$ .

$5 \cdot 4 \sin (\frac{π}{3}) + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.3

Multiply $5$ by $4$ .

$20 \sin (\frac{π}{3}) + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.4

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$20 \frac{\sqrt{3}}{2} + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $20$ .

$2 (10) \frac{\sqrt{3}}{2} + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.5.2

Cancel the common factor.

$2 \cdot 10 \frac{\sqrt{3}}{2} + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.5.3

Rewrite the expression.

$10 \sqrt{3} + 5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.6

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.6.1

Add parentheses.

$10 \sqrt{3} + 5 (\cos (\frac{π}{3}) \tan (\frac{π}{3}))$

Step 1.6.2

Reorder $\cos (\frac{π}{3})$ and $\tan (\frac{π}{3})$ .

$10 \sqrt{3} + 5 (\tan (\frac{π}{3}) \cos (\frac{π}{3}))$

Step 1.6.3

Rewrite $5 \cos (\frac{π}{3}) \tan (\frac{π}{3})$ in terms of sines and cosines.

$10 \sqrt{3} + 5 (\frac{\sin (\frac{π}{3})}{\cos (\frac{π}{3})} \cos (\frac{π}{3}))$

Step 1.6.4

Cancel the common factors.

$10 \sqrt{3} + 5 \sin (\frac{π}{3})$

Step 1.7

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$10 \sqrt{3} + 5 \frac{\sqrt{3}}{2}$

Step 1.8

Combine $5$ and $\frac{\sqrt{3}}{2}$ .

$10 \sqrt{3} + \frac{5 \sqrt{3}}{2}$

Step 2

To write $10 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$10 \sqrt{3} \cdot \frac{2}{2} + \frac{5 \sqrt{3}}{2}$

Step 3

Combine fractions.

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Step 3.1

Combine $10 \sqrt{3}$ and $\frac{2}{2}$ .

$\frac{10 \sqrt{3} \cdot 2}{2} + \frac{5 \sqrt{3}}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{10 \sqrt{3} \cdot 2 + 5 \sqrt{3}}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $10$ .

$\frac{20 \sqrt{3} + 5 \sqrt{3}}{2}$

Step 4.2

Add $20 \sqrt{3}$ and $5 \sqrt{3}$ .

$\frac{25 \sqrt{3}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{25 \sqrt{3}}{2}$

Decimal Form:

$21.65063509 \dots$